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MATHEMATICAL TRIPOS Part II
Friday, 4 June, 2010 9:00 am to 12:00 pm
PAPER 4
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most six questions from Section I and any number of questions from Section II.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in bundles, marked A, B, C,.. ., J according to the code letter affixed to each question. Include in the same bundle all questions from Sections I and II with the same code letter.
Attach a completed gold cover sheet to each bundle.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS
Gold cover sheet Green master cover sheet
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
SECTION I
1G Number Theory Let p be a prime number, and put
ak = kp , Nk = a (^) kp − 1 (k = 1, 2 , ... ).
Prove that ak has exact order p modulo Nk for all k > 1 , and deduce that Nk must be divisible by a prime q with q ≡ 1 (mod p). By making a suitable choice of k, prove that there are infinitely many primes q with q ≡ 1 (mod p).
2F Topics in Analysis Find explicitly a polynomial p of degree 6 3 such that
sup x∈[− 1 ,1]
| x^4 − p(x)| 6 sup x∈[− 1 ,1]
| x^4 − q(x)|
for every polynomial q of degree 6 3. Justify your answer.
3F Geometry of Group Actions Define loxodromic transformations and explain how to determine when a M¨obius transformation
T : z 7 →
az + b cz + d with ad − bc = 1
is loxodromic.
Show that any M¨obius transformation that maps a disc ∆ onto itself cannot be loxodromic.
4H Coding and Cryptography What is the discrete logarithm problem?
Describe the Diffie–Hellman key exchange system for two people. What is the connection with the discrete logarithm problem? Why might one use this scheme rather than just a public key system or a classical (pre-1960) coding system?
Extend the Diffie–Hellman system to n people using n(n − 1) transmitted numbers.
Part II, Paper 4
6A Mathematical Biology A concentration u(x, t) obeys the differential equation
∂u ∂t = Duxx + f (u) ,
in the domain 0 6 x 6 L , with boundary conditions u(0, t) = u(L, t) = 0 and initial condition u(x, 0) = u 0 (x), and where D is a positive constant. Assume f (0) = 0 and f ′(0) > 0. Linearising the dynamics around u = 0, and representing u(x, t) as a suitable Fourier expansion, show that the condition for the linear stability of u = 0 can be expressed as the following condition on the domain length
L < π
[
D
f ′(0)
] 1 / 2
7D Dynamical Systems Consider the 2-dimensional flow
x˙ = y +
x
1 − 2 x^2 − 2 y^2
, y˙ = −x +
y
1 − x^2 − y^2
Use the Poincar´e–Bendixson theorem, which should be stated carefully, to obtain a domain D in the xy-plane, within which there is at least one periodic orbit.
8E Further Complex Methods The hypergeometric function F (a, b; c; z) can be expressed in the form
F (a, b; c; z) =
Γ(c) Γ(b)Γ(c − b)
0
tb−^1 (1 − t)c−b−^1 (1 − tz)−a^ dt ,
for appropriate restrictions on c, b, z.
Express the following integral in terms of a combination of hypergeometric functions
I(u, A) =
∫ π 2
− π 2
eit(u+1) eit^ + iA
dt , |A| > 1.
[You may use without proof that Γ(z + 1) = zΓ(z) .]
Part II, Paper 4
9D Classical Dynamics A system with one degree of freedom has Lagrangian L(q, q˙). Define the canonical momentum p and the energy E. Show that E is constant along any classical path.
Consider a classical path qc(t) with the boundary–value data
qc(0) = qI , qc(T ) = qF , T > 0.
Define the action Sc(qI , qF , T ) of the path. Show that the total derivative dSc/dT along the classical path obeys dSc dT
= L.
Using Lagrange’s equations, or otherwise, deduce that
∂Sc ∂qF = pF , ∂Sc ∂T
= −E ,
where pF is the final momentum.
10D Cosmology The linearised equation for the growth of density perturbations, δk, in an isotropic and homogenous universe is
¨δk + 2 a˙ a
δ˙k +
cs^2 k^2 a 2
− 4 π Gρ
δk = 0 ,
where ρ is the density of matter, cs the sound speed, cs^2 = dP/dρ , and k is the comoving wavevector and a(t) is the scale factor of the universe.
What is the Jean’s length? Discuss its significance for the growth of perturbations.
Consider a universe filled with pressure-free matter with a(t) = (t/t 0 )^2 /^3. Compute the resulting equation for the growth of density perturbations. Show that your equation has growing and decaying modes and comment briefly on the significance of this fact.
Part II, Paper 4 [TURN OVER
13J Statistical Modelling Every day, Barney the darts player comes to our laboratory. We record his facial expression, which can be either “mad”, “weird” or “relaxed”, as well as how many units of beer he has drunk that day. Each day he tries a hundred times to hit the bull’s-eye, and we write down how often he succeeds. The data look like this:
Day Beer Expression BullsEye 1 3 Mad 30 2 3 Mad 32 ... ... ... ... 60 2 Mad 37 61 4 Weird 30 ... ... ... ... 110 4 Weird 28 111 2 Relaxed 35 ... ... ... ... 150 3 Relaxed 31
Write down a reasonable model for Y 1 ,... , Yn, where n = 150 and where Yi is the number of times Barney has hit bull’s-eye on the ith day. Explain briefly why we may wish initially to include interactions between the variables. Write the R code to fit your model.
The scientist of the above story fitted her own generalized linear model, and subsequently obtained the following summary (abbreviated):
summary(barney) [...]
Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.37258 0.05388 -6.916 4.66e-12 *** Beer -0.09055 0.01595 -5.676 1.38e-08 *** ExpressionWeird -0.10005 0.08044 -1.244 0. ExpressionRelaxed 0.29881 0.08268 3.614 0.000301 *** Beer:ExpressionWeird 0.03666 0.02364 1.551 0. Beer:ExpressionRelaxed -0.07697 0.02845 -2.705 0.006825 **
[...]
Why are ExpressionMad and Beer:ExpressionMad not listed? Suppose on a particular day, Barney’s facial expression is weird, and he drank three units of beer. Give the linear predictor in the scientist’s model for this day.
Based on the summary, how could you improve your model? How could one fit this new model in R (without modifying the data file)?
Part II, Paper 4 [TURN OVER
14D Dynamical Systems Let I = [ 0, 1 ] and consider continuous maps F : I → I. Give an informal outline description of the two different bifurcations of fixed points of F that can occur.
Illustrate your discussion by considering in detail the logistic map
F (x) = μ x (1 − x) ,
for μ ∈ (0, 1 +
6 ].
Describe qualitatively what happens for μ ∈ (1 +
6 , 4].
[You may assume without proof that
x − F 2 (x) = x (μ x − μ + 1) ( μ^2 x^2 − μ(μ + 1) x + μ + 1 ). ]
15D Classical Dynamics A system is described by the Hamiltonian H(q, p). Define the Poisson bracket {f, g} of two functions f (q, p, t), g(q, p, t), and show from Hamilton’s equations that
df dt
= {f, H} + ∂f ∂t
Consider the Hamiltonian
H =
(p^2 + ω^2 q^2 ) ,
and define a = (p − iωq)/(2ω)^1 /^2 , a∗^ = (p + iωq)/(2ω)^1 /^2 , where i =
−1. Evaluate {a, a} and {a, a∗}, and show that {a, H} = −iωa and {a∗, H} = iωa∗. Show further that, when f (q, p, t) is regarded as a function of the independent complex variables a, a∗^ and of t, one has
df dt
= iω
a∗^ ∂f ∂a∗^
− a ∂f ∂a
∂f ∂t
Deduce that both log a∗^ − iωt and log a + iωt are constant during the motion.
Part II, Paper 4
19F Representation Theory Define the circle group U (1). Give a complete list of the irreducible representations of U (1).
Define the spin group G = SU (2), and explain briefly why it is homeomorphic to the unit 3-sphere in R^4. Identify the conjugacy classes of G and describe the classification of the irreducible representations of G. Identify the characters afforded by the irreducible representations. You need not give detailed proofs but you should define all the terms you use.
Let G act on the space M 3 (C) of 3 × 3 complex matrices by conjugation, where A ∈ SU (2) acts by A : M 7 → A 1 M A 1 −^1 ,
in which A 1 denotes the 3 × 3 block diagonal matrix
A 0
. Show that this gives a
representation of G and decompose it into irreducibles.
20G Number Fields Suppose that α is a zero of x 3 − x + 3 and that K = Q(α). Show that [K : Q] = 3. Show that OK , the ring of integers in K, is OK = Z [α].
[You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of x 3 + px + q is − 4 p^3 − 27 q^2 .]
21H Algebraic Topology State the Snake Lemma. Explain how to define the boundary map which appears in it, and check that it is well-defined. Derive the Mayer-Vietoris sequence from the Snake Lemma.
Given a chain complex C, let A ⊂ C be the span of all elements in C with grading greater than or equal to n , and let B ⊂ C be the span of all elements in C with grading less than n. Give a short exact sequence of chain complexes relating A , B, and C. What is the boundary map in the corresponding long exact sequence?
Part II, Paper 4
22H Linear Analysis Let X be a Banach space.
a) What does it mean for a bounded linear map T : X → X to be compact?
b) Let B(X) be the Banach space of all bounded linear maps S : X → X. Let B 0 (X) be the subset of B(X) consisting of all compact operators. Show that B 0 (X) is a closed subspace of B(X). Show that, if S ∈ B(X) and T ∈ B 0 (X), then ST, T S ∈ B 0 (X).
c) Let
X = ℓ^2 =
x = (x 1 , x 2 ,... ) : xj ∈ C and ‖x‖^22 =
∑^ ∞
j=
|xj |^2 < ∞
and T : X → X be defined by (T x)k =
xk+ k + 1
Is T compact? What is the spectrum of T? Explain your answers.
23G Algebraic Geometry Let E ⊆ P^2 be the projective curve obtained from the affine curve y 2 = (x − λ 1 )(x − λ 2 )(x − λ 3 ), where the λ (^) i are distinct and λ 1 λ 2 λ 3 6 = 0.
(i) Show there is a unique point at infinity, P∞.
(ii) Compute div(x), div(y).
(iii) Show L (P∞) = k.
(iv) Compute l(nP∞) for all n. [You may not use the Riemann–Roch theorem.]
Part II, Paper 4 [TURN OVER
27J Principles of Statistics Define completeness and bounded completeness of a statistic T in a statistical experiment.
Random variables X 1 , X 2 , X 3 are generated as Xi = Θ^1 /^2 Z + (1 − Θ)^1 /^2 Yi , where Z, Y 1 , Y 2 , Y 3 are independently standard normal N (0, 1), and the parameter Θ takes values in (0, 1). What is the joint distribution of (X 1 , X 2 , X 3 ) when Θ = θ? Write down its density function, and show that a minimal sufficient statistic for Θ based on (X 1 , X 2 , X 3 ) is T = (T 1 , T 2 ) := (
i=1 X 2 i ,^ (
i=1 Xi)
[Hint: You may use that if I is the n × n identity matrix and J is the n × n matrix all of whose entries are 1, then aI + bJ has determinant an−^1 (a + nb), and inverse cI + dJ with c = 1/a , d = −b/(a(a + nb)).]
What is Eθ(T 1 )? Is T complete for Θ?
Let S := Prob(X 12 6 1 | T ). Show that Eθ(S) is a positive constant c which does not depend on θ, but that S is not identically equal to c. Is T boundedly complete for Θ?
Part II, Paper 4 [TURN OVER
28J Optimization and Control Dr Seuss’ wealth xt at time t evolves as
dx dt = rxt + ℓt − ct,
where r > 0 is the rate of interest earned, ℓt is his intensity of working (0 6 ℓ 6 1), and ct is his rate of consumption. His initial wealth x 0 > 0 is given, and his objective is to maximize (^) ∫ (^) T
0
U (ct, ℓt) dt,
where U (c, ℓ) = cα(1 − ℓ)β^ , and T is the (fixed) time his contract expires. The constants α and β satisfy the inequalities 0 < α < 1, 0 < β < 1, and α + β > 1. At all times, ct must be non-negative, and his final wealth xT must be non-negative. Establish the following properties of the optimal solution (x∗, c∗, ℓ∗):
(i) βc∗ t = α(1 − ℓ∗ t );
(ii) c∗ t ∝ e−γrt, where γ ≡ (β − 1 + α)−^1 ;
(iii) x∗ t = Aert^ + Be−γrt^ − r−^1 for some constants A and B.
Hence deduce that the optimal wealth is
x∗ t = (1 − e−γrT^ (1 + rx 0 ))ert^ + ((1 + rx 0 )erT^ − 1)e−γrt r(erT^ − e−γrT^ )
r
Part II, Paper 4
30E Partial Differential Equations a) Solve the Dirichlet problem for the Laplace equation in a disc in R^2
∆u = 0 in G = {x^2 + y^2 < R^2 } ⊆ R^2 , R > 0 , u = uD on ∂G ,
using polar coordinates (r, ϕ) and separation of variables, u(x, y) = R(r)Θ(ϕ). Then use the ansatz R(r) = rα^ for the radial function.
b) Solve the Dirichlet problem for the Laplace equation in a square in R^2
∆u = 0 in G = [0, a] × [0, a] , u(x, 0) = f 1 (x) , u(x, a) = f 2 (x) , u(0, y) = f 3 (y) , u(a, y) = f 4 (y).
31C Asymptotic Methods (a) Consider for λ > 0 the Laplace type integral
I(λ) =
∫ (^) b
a
f (t) e−λφ(t)^ dt ,
for some finite a, b ∈ R and smooth, real-valued functions f (t), φ(t). Assume that the function φ(t) has a single minimum at t = c with a < c < b. Give an account of Laplace’s method for finding the leading order asymptotic behaviour of I(λ) as λ → ∞ and briefly discuss the difference if instead c = a or c = b, i.e. when the minimum is attained at the boundary.
(b) Determine the leading order asymptotic behaviour of
I(λ) =
− 2
cos t e−λt 2 dt , (∗)
as λ → ∞.
(c) Determine also the leading order asymptotic behaviour when cos t is replaced by sin t in (∗).
Part II, Paper 4
32C Principles of Quantum Mechanics The Hamiltonian for a quantum system in the Schr¨odinger picture is
H 0 + V (t) ,
where H 0 is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.
Let |a〉 and |b〉 be orthonormal eigenstates of H 0 with eigenvalues Ea and Eb respectively. Assume V (t) = 0 for t 6 0. Show that if the system is initially, at t = 0, in the state |a〉 then the probability of measuring it to be the state |b〉 after a time t is
1 ℏ^2
∫ (^) t
0
dt′〈b|V (t′)|a〉e i(Eb^ −Ea)t
′/ℏ
2 (∗)
to order V (t)^2.
Suppose a system has a basis of just two orthonormal states | 1 〉 and | 2 〉, with respect to which H 0 = E I , V (t) = vt σ 1 , t > 0 , where I =
, σ 1 =
Use (∗) to calculate the probability of a transition from state | 1 〉 to state | 2 〉 after a time t to order v^2. Show that the time dependent Schr¨odinger equation has a solution
|ψ(t)〉 = exp
i ℏ
Et I + 12 vt^2 σ 1
|ψ(0)〉.
Calculate the transition probability exactly. Hence find the condition for the order v^2 approximation to be valid.
Part II, Paper 4 [TURN OVER
34C Statistical Physics
(i) Let ρi be the probability that a system is in a state labelled by i with Ni particles and energy Ei. Define s(ρi) = −k
i
ρi log ρi.
s(ρi) has a maximum, consistent with a fixed mean total number of particles N , mean total energy E and
i ρi^ = 1, when^ ρi^ = ¯ρi. Let^ S(E, N^ ) =^ s(¯ρi) and show that ∂S ∂E
T
∂S
∂N
μ T
where T may be identified with the temperature and μ with the chemical potential.
(ii) For two weakly coupled systems 1,2 then ρi,j = ρ 1 ,i ρ 2 ,j and Ei,j = E 1 ,i + E 2 ,j , Ni,j = N 1 ,i + N 2 ,j. Show that S(E, N ) = S 1 (E 1 , N 1 ) + S 2 (E 2 , N 2 ) where, if S(E, N ) is stationary under variations in E 1 , E 2 and N 1 , N 2 for E = E 1 + E 2 , N = N 1 + N 2 fixed, we must have T 1 = T 2 , μ 1 = μ 2.
(iii) Define the grand partition function Z(T, μ) for the system in (i) and show that
k log Z = S −
T
E +
μ T
N , S =
∂T
kT log Z
(iv) For a system with single particle energy levels ǫr the possible states are labelled by i = {nr : nr = 0, 1 }, where Ni =
r nr,^ Ei^ =^
r nrǫr^ and^
i =^
r
nr =0, 1. Show that ρ¯i =
r
e−nr^ (ǫr^ −μ)/kT 1 + e−(ǫr^ −μ)/kT^
Calculate ¯nr. How is this related to a free fermion gas?
35B Electrodynamics In a superconductor the number density of charge carriers of charge q is ns. Suppose that there is a time-independent magnetic field described by the three-vector potential A.
Derive an expression for the superconducting current.
Explain how your answer is gauge invariant.
Suppose that for z < 0 there is a constant magnetic field B 0 in a vacuum and, for z > 0, there is a uniform superconductor. Derive the magnetic field for z > 0.
Part II, Paper 4 [TURN OVER
36B General Relativity The Schwarzschild line element is given by
ds^2 = −F dt^2 + F −^1 dr^2 + r^2 (dθ^2 + sin^2 θ dφ^2 ) ,
where F = 1 − rs/r and rs is the Schwarzschild radius. Obtain the equation of geodesic motion of photons moving in the equatorial plane, θ = π/2, in the form
( (^) dr dτ
= E^2 −
h^2 F r^2
where τ is proper time, and E and h are constants whose physical significance should be indicated briefly.
Defining u = 1/r show that light rays are determined by ( (^) du dφ
b
− u^2 + rs u^3 ,
where b = h/E and rs may be taken to be small. Show that, to zeroth order in rs, a light ray is a straight line passing at distance b from the origin. Show that, to first order in rs, the light ray is deflected through an angle 2rs/b. Comment briefly on some observational evidence for the result.
Part II, Paper 4
